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                <h1 id="title" titleSize="">
                    Graph
                </h1>
            
            <h1 id="motivation--definition">Motivation &amp; Definition</h1>
<p>The concept of a <em>graph</em> arose as an abstraction of a typical situation in math where one has a set of ‘objects’ $V$ on which ‘connections’ between those objects can be established. The precise definition of a graph differs, depending on who you ask $-$ before embarking on the general definition, let us first define the simplest type of graphs, which will (unless otherwise stated) be our focus in these notes.</p>
<div class="env envDef" id=""><img class="icon noSelect listenDark" src="https://zhaoshenzhai.github.io/mathwiki/css/fa/definition.svg"><b class="envTitle">Definition. </b><p>A <em>(simple loopless undirected) graph</em> is a pair $G\coloneqq(V,E)$, where $V$ is a set and $E\subseteq V^2$ is a symmetric irreflexive binary relation. We write $uv\coloneqq(u,v)\eqqcolon e\in E$, in which case $u$ (and also $v$) are <em>incident</em> to $e$, and $u,v$ are <em>adjacent</em>.</p>
<br>
<p>  A function $f:(V,E)\to(V&rsquo;,E&rsquo;)$ between graphs is a <em>graph homomorphism</em> if $uv\in E$ implies $f(u)f(v)\in E&rsquo;$.</p>
</div>

<p>With graph homomorphisms as morphisms, simple graphs form a <a href=https://zhaoshenzhai.github.io/mathwiki/category.md class="internalLink references" title="Category" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/category.md&#34;, {&#34;Date&#34;:&#34;2024-05-22T21:52:00-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-05-22T21:52:00-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-05-22T21:52:00-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Category&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/category&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-05-22T21:52:00-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-05-22T21:52:00-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-05-22T21:52:00-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Category&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/category&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/category.md&#34;, {&#34;Date&#34;:&#34;2024-05-22T21:52:00-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-05-22T21:52:00-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-05-22T21:52:00-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Category&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/category&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">category</a> $\catsgrph$. Note that graphs are also models of the $\mc{L}$-<a href=https://zhaoshenzhai.github.io/mathwiki/first_order_theory.md class="internalLink references ghostLink" title="theory" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/first_order_theory.md&#34;, &#34;nopPage&#34;);" onmouseleave="clearPreviewSide(&#34;nopPage&#34;);" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/first_order_theory.md&#34;, &#34;nopPage&#34;);">theory</a> $T$ of graphs (where $\mc{L}\coloneqq\l\{E\r\}$ contains a binary relation symbol) stating that $E$ is symmetric and irreflexive, so $\catsgrph=\cathom_\mc{L}$.</p>
<h2 id="basic-constructions">Basic Constructions</h2>
<p>Fix a graph $G\coloneqq(V,E)$. A <em>subgraph</em> of a graph $G\coloneqq(V,E)$ is a <a href=https://zhaoshenzhai.github.io/mathwiki/substructure.md class="internalLink references ghostLink" title="substructure" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/substructure.md&#34;, &#34;nopPage&#34;);" onmouseleave="clearPreviewSide(&#34;nopPage&#34;);" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/substructure.md&#34;, &#34;nopPage&#34;);">substructure</a> of $G$ in $\cathom_\mc{L}$, which is a graph $G&rsquo;\coloneqq(V&rsquo;,E&rsquo;)$ with $V&rsquo;\subseteq V$ and $E&rsquo;\subseteq E$. On the other hand, considering the category $\catemb_\mc{L}$ of graphs with embeddings as morphisms instead gives us an <em>induced subgraph</em> of $G$, which is a subgraph $G&rsquo;\coloneqq(V&rsquo;,E&rsquo;)$ of $G$ such that $E&rsquo;=V&rsquo;^2\cap E$.</p>
<br>
<p>  Since <a href=https://zhaoshenzhai.github.io/mathwiki/limits_category_theory.md class="internalLink references" title="Limits &amp; Colimits" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/limits_category_theory.md&#34;, {&#34;Date&#34;:&#34;2024-05-29T20:16:10-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-05-29T20:16:10-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-05-29T20:16:10-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Limits \u0026 Colimits&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/limits_category_theory&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-05-29T20:16:10-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-05-29T20:16:10-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-05-29T20:16:10-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Limits \u0026 Colimits&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/limits_category_theory&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/limits_category_theory.md&#34;, {&#34;Date&#34;:&#34;2024-05-29T20:16:10-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-05-29T20:16:10-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-05-29T20:16:10-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Limits \u0026 Colimits&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/limits_category_theory&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">coproducts</a> in $\catsgrph$ are given by the disjoint union of their underlying vertices and edges, we see that $G$ decomposes as a coproduct of its <a href=https://zhaoshenzhai.github.io/mathwiki/connectivity_graphs.md/#connected_components class="internalLink constructions" title="Connectivity in Graphs" mathLink="" secID="connected_components" secDisplay="Connected Components" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/connectivity_graphs.md/#connected_components&#34;, {&#34;Date&#34;:&#34;2024-09-05T22:37:16-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-09-05T22:37:16-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-09-05T22:37:16-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Connectivity in Graphs&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/connectivity_graphs&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-09-05T22:37:16-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-09-05T22:37:16-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-09-05T22:37:16-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Connectivity in Graphs&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/connectivity_graphs&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/connectivity_graphs.md/#connected_components&#34;, {&#34;Date&#34;:&#34;2024-09-05T22:37:16-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-09-05T22:37:16-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-09-05T22:37:16-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Connectivity in Graphs&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/connectivity_graphs&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">connected</a> subgraphs. Thus, we will assume here that $G$ is itself connected. <span style="color:red"><strong>TODO</strong>.</span></p>
<div class="space"></div>
<p>  The next section extends the definition to other ‘graph-like’ objects. We found that the most systematic way to talk about these variations is by viewing graphs as <a href=https://zhaoshenzhai.github.io/mathwiki/presheaf.md class="internalLink references ghostLink" title="presheaves" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/presheaf.md&#34;, &#34;nopPage&#34;);" onmouseleave="clearPreviewSide(&#34;nopPage&#34;);" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/presheaf.md&#34;, &#34;nopPage&#34;);">presheaves</a>, but this language is not strictly necessary.</p>
<h2 class="hide">Graphs as Presheaves</h2>
<p>Consider a tuple $G\coloneqq(V,E,s,t)$, where $s,t:E\tto V$ are parallel maps between sets $V,E$. This setup is equivalent to a presheaf $\cat{C}^\textrm{op}\to\catset$, where $\cat{C}$ is the <a href=https://zhaoshenzhai.github.io/mathwiki/free_category.md class="internalLink references ghostLink" title="free category" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/free_category.md&#34;, &#34;nopPage&#34;);" onmouseleave="clearPreviewSide(&#34;nopPage&#34;);" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/free_category.md&#34;, &#34;nopPage&#34;);">free category</a> generated by $E\tto V$,</p>
<br>
<p>  <span style="color:red"><strong>TODO</strong> (see <a href="https://mathoverflow.net/questions/321897/is-mathrmgraph-cartesian-closed?noredirect=1&amp;lq=1)">https://mathoverflow.net/questions/321897/is-mathrmgraph-cartesian-closed?noredirect=1&amp;lq=1)</a>.</span></p>
<h1 id="techniques-and-paradigms">Techniques and Paradigms</h1>
<h2 id="hahahugoshortcode30s8hbhb-theory"><a href=https://zhaoshenzhai.github.io/mathwiki/finite_graph.md class="internalLink types" title="Finite Graph" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/finite_graph.md&#34;, {&#34;Date&#34;:&#34;2024-09-06T23:44:59-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-09-06T23:44:59-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-09-06T23:44:59-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Finite Graph&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/finite_graph&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-09-06T23:44:59-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-09-06T23:44:59-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-09-06T23:44:59-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Finite Graph&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/finite_graph&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/finite_graph.md&#34;, {&#34;Date&#34;:&#34;2024-09-06T23:44:59-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-09-06T23:44:59-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-09-06T23:44:59-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Finite Graph&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/finite_graph&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">Finite Graph</a> Theory</h2>
<h2 id="infinite-graph-theory">Infinite Graph Theory</h2>
<h1 id="applications-and-generalizations">Applications and Generalizations</h1>
<h2 id="geometric-group-theory">Geometric Group Theory</h2>
<h2 id="simplicial-complexes">Simplicial Complexes</h2>


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                September 5, 2024 | Zhaoshen Zhai

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